Prove that the product of 2π

Question

Prove that the product of  2π
  and  3/4
  is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1–2 sentences.(2 points)

GPT-4o mini · Answer

To find the product of $ 2\pi $ and $ \frac{3}{4} $, we calculate: $$ 2\pi \cdot \frac{3}{4} = \frac{3}{2} \pi $$ The number $ \pi $ is known to be irrational, and the product of a non-zero rational number ($ \frac{3}{2} $) and an irrational number ($ \pi $) is also irrational. Therefore, $ \frac{3}{2} \pi $ is irrational. In conclusion, $ \frac{3}{2} \pi $ is irrational because it is the product of a non-zero rational number and an irrational number.